| L(s) = 1 | − 4.80·2-s + 2.56·3-s + 15.0·4-s + 6.34·5-s − 12.3·6-s − 6.17·7-s − 34.0·8-s − 20.4·9-s − 30.4·10-s + 33.6·11-s + 38.6·12-s − 34.3·13-s + 29.6·14-s + 16.2·15-s + 42.7·16-s + 105.·17-s + 98.0·18-s − 14.3·19-s + 95.6·20-s − 15.8·21-s − 161.·22-s − 160.·23-s − 87.2·24-s − 84.7·25-s + 165.·26-s − 121.·27-s − 93.1·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.493·3-s + 1.88·4-s + 0.567·5-s − 0.838·6-s − 0.333·7-s − 1.50·8-s − 0.756·9-s − 0.963·10-s + 0.921·11-s + 0.930·12-s − 0.733·13-s + 0.566·14-s + 0.280·15-s + 0.668·16-s + 1.51·17-s + 1.28·18-s − 0.172·19-s + 1.06·20-s − 0.164·21-s − 1.56·22-s − 1.45·23-s − 0.742·24-s − 0.678·25-s + 1.24·26-s − 0.867·27-s − 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 4.80T + 8T^{2} \) |
| 3 | \( 1 - 2.56T + 27T^{2} \) |
| 5 | \( 1 - 6.34T + 125T^{2} \) |
| 7 | \( 1 + 6.17T + 343T^{2} \) |
| 11 | \( 1 - 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 0.100T + 2.97e4T^{2} \) |
| 37 | \( 1 - 369.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 450.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 288.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 439.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 16.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 764.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 415.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 998.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 160.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 374.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 706.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.557913047649090499208928054848, −7.911023282407672548708004964167, −7.31574925449418198235567469903, −6.20441254146523374733221326462, −5.75458641663811224810756184168, −4.13540712316786527769393915232, −2.93174091892748115165401749455, −2.15910813545392618080103792104, −1.16030584969930639244751217619, 0,
1.16030584969930639244751217619, 2.15910813545392618080103792104, 2.93174091892748115165401749455, 4.13540712316786527769393915232, 5.75458641663811224810756184168, 6.20441254146523374733221326462, 7.31574925449418198235567469903, 7.911023282407672548708004964167, 8.557913047649090499208928054848
